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PRICING CREDIT RISK DERIVATIVES 1. Introduction Credit

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PRICING CREDIT RISK DERIVATIVES 1. Introduction Credit PRICING CREDIT RISK DERIVATIVES PHILIPP J. SCHONBUCHER¨ Abstract. In this paper the pricing of several credit risk derivatives is discussed in an intensity-based framework with both risk-free and defaultable interest rates stochas- tic and possibly correlated. Credit spread puts and exchange options serve as main examples. The differences to standard interest rate derivatives are analysed. Where possible closed-form solutions are given. 1. Introduction Credit derivatives are derivative securities whose payoff depends on the credit quality of a certain issuer. This credit quality can be measured by the credit rating of the issuer or by the yield spread of his bonds over the yield of a comparable default-free bond. In this paper we will concentrate on the latter case. Credit Risk Derivatives (CRD) have enjoyed much attention and publicity in the deriva- tives world in recent years. They have been hailed as the major new risk management tool for managing credit exposures, as the most important new market in fixed in- come derivatives and as a breakthrough in loan risk management. This praise is at least to some part justified: Credit risk derivatives can make large and important risks tradeable. They form an important step towards market completion and efficient risk allocation, they can help bridge the traditional market segmentation between corporate loan and bond markets. Despite their large potential practical importance, there have been very few works specifically on the pricing and hedging of credit risk derivatives. The majority of papers is concerned with the pricing of defaultable bonds which is a necessary prerequisite for credit derivatives pricing. The leading recovery model in this paper is the fractional recovery / multiple defaults model (see Duffie and Singelton [7, 8] and also Sch¨onbucher [17, 18]). Applying this model to credit derivatives will yield important insights into the subtler points of the model used. The choice of model is not essential to the main point of the paper which is to show the methodology that has to be used in an Date: July 1997, this version January 1998. Financial support by the DFG, SFB 303 at Bonn University and by the DAAD are gratefully acknowledged. I would like to thank Erik Schl¨ogland David Lando for helpful discussions. Comments and suggestions to this paper are very welcome. This is an early draft. All errors are of course my own. Author’s address: Department of Statistics, Faculty of Economics, University of Bonn, Adenauerallee 24-42, 53113 Bonn, Germany. email: schonbuc@addi.finasto.uni-bonn.de. 1 2 PHILIPP J. SCHONBUCHER¨ intensity-based credit spread model. It is easy to transfer most of the results to other intensity-based models like e.g. the models by Jarrow and Turnbull [12], Lando [14, 13], Madan and Unal [16] or Artzner and Delbaen [1], in fact it is demonstrated that under some circumstances it may be more appropriate to model the recoveries following one of these models. In the following section this model will be recapitulated, followed by a presentation of the credit derivatives that will be priced. As the vast majority of credit derivatives are OTC derivatives, no standard specification has evolved for second generation derivatives yet. Therefore we chose an array of typical and/or natural specifications to exemplify typical cases, including default swaps, straight put options on defaultable bonds, credit spread puts with different specifications for the treatment of default events, and default digital puts. Then we define the specification of the stochastic processes. We chose a multifactor Cox Ingersoll Ross (CIR) [4] square-root diffusion model both for the credit spread and for the default-free short rate. This setup can allow for arbitrary correlation between credit spreads and the risk-free term structure of interest rates. To keep the results as flexible as possible we used this specification only in the last step of the calculation of the prices, and we also give the expectations that have to be calculated if another model is used. This specification is used in the next section to price the credit risk derivatives that were introduced before. The pricing formulae are expressed in terms of the multivariate noncentral chi-squared distribution function, using results from Jamshidian [11]. The conclusion sums up the main results of the paper and points out the consequences that these results have for the further development of credit spread models. 2. Defaultable Bond Pricing with Cox Processes The model used here is an extension of the Duffie-Singleton [7] model to multiple de- faults. More details to the model can be found in Sch¨onbucher [17, 18]. The new feature of this model is that a default does not lead to a liquidation but a reorganisation of the issuer: defaulted bonds lose a fraction q of their face value and continue to trade. This feature enables us to consider European-type payoffs in our derivatives without necessarily needing to specify a payoff of the derivative at default (although we will consider this case, too). All dynamics that we introduce are understood to be specified with respect to a common martingale measure. First we will have to introduce some notation: 2.0.1. The default-free term structure. The term structure of default-free bond prices is given by the time t prices P (t, T ) of the default-free zero coupon bonds with maturity T t. It is well-known that under the martingale measure ≥ R T r(s)ds (1) P (t, T ) = Et h e− t i , PRICING CREDIT RISK DERIVATIVES 3 where r denotes the default-free short rate. We will specify the dynamics of r later on. The discount factor βt,T over the interval [t, T ] is defined as R T r(s)ds (2) βt,T = e− t . 2.0.2. The defaultable bond prices. Defaultable bonds are the underlying securities to most credit risk derivatives. The time t price of a defaultable zero coupon bond with maturity T is denoted 1 with P 0(t, T ). The defaults themselves are modeled as follows: There is an increasing sequence of stopping times τi i IN that define the times of default. At each default the defaultable bond’s face{ value} ∈ is reduced by a factor q, where q may be a random variable itself. This means that a defaultable zero coupon bond’s final payoff is the product Q(T ) := (1 qi)(3) Y − τi T ≤ of the face value reductions after all defaults until the maturity T of the defaultable bond. The loss quotas qi can be random variables drawn from a distribution K(dq) at time τi, but for the first calculations we will assume qi = q to be constant. To simplify notation the time of the first default will often be referred to with τ := τ1. We assume that the times of default τi are generated by a Cox process. Intuitively, a Cox Process is defined as a Poisson process with stochastic intensity h (see Lando [13]). Formally the definition is: Definition 1. The default counting process ∞ (4) N(t) := max i τi t = 1 τi t X { ≤ } { | ≤ } i=1 is called a Cox process, if there is a nonnegative stochastic process h(t) (called the intensity of the Cox process), such that given the realization h(t) t>0 of the intensity, N(t) is a time-inhomogeneous Poisson process with intensity{ h(t)}.{ } This means that the probability of having exactly n jumps, given the realisation of h, is n 1 T T (5) Pr N(T ) N(t) = n h(s) T s t = Z h(s)ds exp Z h(s)ds , − | { }{ ≥ ≥ } n! t {− t } and the probability of having n jumps (without knowledge of the realisation of h) is found by conditioning on the realisation of h within an outer expectation operator: Pr [ N(T ) N(t) = n ] = Et Pr N(T ) N(t) = n h(s) T s t { ≥ ≥ } − − n | { } 1 T T (6) = Et Z h(s)ds exp Z h(s)ds , n! t {− t } 1Defaultable quantities usually carry a dash (’). 4 PHILIPP J. SCHONBUCHER¨ Specifically, the survival probability Ph(t, T ) of survival until time T as viewed from time t is R T h(s)ds (7) Ph(t, T ) = E 1 τ>T = E e− t . { } h i This application of the law of iterated expectations will prove extremely useful later on, it was first used in a credit risk context by Lando [13]. The specification of the default trigger process as a Cox process precludes a dependence of the default intensity on previous defaults and also ensures totally inaccessible stopping times τi as times of default. Apart from this it allows rich dynamics of the intensity process, specifically, we can reach stochastic credit spreads. Equations (5) and (6) will be used frequently later on. It is now easily seen (using the iterated expectations, see also Duffie and Singleton and Sch¨onbucher [7, 17, 18]) that in this setup the price of a defaultable zero coupon bond is given by R T r0(s)ds (8) P 0(t, T ) = Q(t)Et h e− t i . The process r0 is called the defaultable short rate r0 and it is defined by (9) r0 = r + hq. Here r is the default-free short rate, h the hazard rate of the defaults and q is the loss quota in default. If q is stochastic then q has to replaced by its (local) expectation e qt = qKt(dq) in equation (9). The similarity of equation (8) to the default-free bond pricingR equation (1) will be used later on.
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